**1. Introduction**

Guided modes in circular waveguides consist of metamaterials [1–13] have been studied in the literature. Many studies of propagation modes in this waveguides with isotropic media [14–17] or double negative metamaterials [18, 19] have been presented in the literature. However, the rigorous study of the dispersion of anisotropic metamaterials in circular waveguides presents a lack in the literature. In this chapter, we present an extension of the rigorous analysis of the propagation of electromagnetic waves in magnetic transverse (TM) and electric transverse (TE) modes in the case of anisotropic circular waveguides, who take account of the spatial distribution of the permittivity and permeability of the medium. In this

structure, the propagation modes are exploited. The effects of anisotropic parameter on cutoff frequencies and dispersion characteristics are discussed. Below the cutoff frequency, the back backward waves can propagate in an anisotropic material. The numerical results with our MATLAB code for TM and TE modes were compared to theoretical predictions. Good agreements have been obtained. We analyzed a waveguide filters filled with partially anisotropic metamaterial using the mode matching (MM) technique based on the Scattering Matrix Approach (SMA) which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. We extended the application of MM technique to the anisotropic material.

This formulation can be a useful tool for engineers of microwave. The metamaterial is largely applied by information technology industries, particularly in the radio frequency devices and microwaves such as the waveguide antennas, the patch antennas, the circulators, the resonators and the filters.

### **2. Formulation**

In the anisotropic diagonal metamaterials medium, the Maxwell equations are expressed as follows

$$
\overrightarrow{\nabla} \times \overrightarrow{E} = -j a \overline{\mu} \,\overrightarrow{H} \tag{1}
$$

*<sup>H</sup><sup>θ</sup>* <sup>¼</sup> *<sup>j</sup> K*2 *c:r*

*Geometry of circular waveguide filled with metamaterial.*

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

*K*2 *<sup>c</sup>:<sup>r</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

*K*2 *<sup>c</sup>:<sup>θ</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup>

> *k*2 <sup>0</sup> <sup>¼</sup> *<sup>ω</sup>*<sup>2</sup>

permittivity and permeability. *kz* is the propagation constant in z-direction.

with

**Figure 1.**

waveguide.

*∂*2 *Hz ∂r*<sup>2</sup> þ

*E*ð Þ *<sup>h</sup>*

**67**

*<sup>r</sup>* <sup>¼</sup> �*jωμ*0*μr<sup>θ</sup> K*2 *<sup>c</sup>:<sup>r</sup>:r*

**2.1 Transverse electric (TE) modes**

1 *r ∂Hz ∂r* þ

tial Eq. (12). *Hz* can be written as follows

*<sup>z</sup>* ¼ *H*<sup>0</sup> sin

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

!

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup> *n:θ*

*H*ð Þ *<sup>h</sup>*

The expressions (5)–(8) become

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

!<sup>2</sup>

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

!

*Jn* is the Bessel function of the first kind of order *n* (*n* = 0, 1, 2, 3, … ).

*H*0*:* cos

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup> *n:θ*

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup> �*ωε*0*εrr*

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

*∂Ez <sup>∂</sup><sup>r</sup>* � *kz r ∂Hz ∂θ* � � (8)

<sup>0</sup>*εrrμr<sup>θ</sup>* � *<sup>k</sup>*<sup>2</sup>

<sup>0</sup>*εrθμrr* � *<sup>k</sup>*<sup>2</sup>

When *E* and *H* are the electric and magnetic field respectively. *ε* and *μ* are the

In this chapter, we study rigorously the TE and TM modes in this anisotropic

From Eq. (1), the differential equation for z-component can be obtained as follows

ffiffiffiffiffiffi *<sup>μ</sup>rz* <sup>p</sup> ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup>

> ffiffiffiffiffiffi *<sup>μ</sup>rz* <sup>p</sup> ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup>

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :r* � �*<sup>e</sup>*

*Jn*

ffiffiffiffiffiffi *<sup>μ</sup>rz* <sup>p</sup> ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup>

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :r* � �*<sup>e</sup>*

�*jkzz*

(14)

*K*ð Þ *<sup>h</sup> c:θ* � �<sup>2</sup>

1 *r*2 *∂*2 *Hz ∂θ*<sup>2</sup> þ

Using the separation of the variables (*r,θ*), the expression of the longitudinal magnetic field *Hz* for the *TEmn* modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differen-

*Jn*

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

!

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *<sup>μ</sup>rr* <sup>p</sup> *n:θ*

*<sup>z</sup>* (9)

*<sup>z</sup>* (10)

*Hz* ¼ 0*:* (12)

�*jkzz* (13)

*ε*0*μ*<sup>0</sup> (11)

$$
\overrightarrow{\nabla} \times \overrightarrow{H} = j o \overline{\overline{\varepsilon}}.\overrightarrow{E} \tag{2}
$$

with

$$
\overline{\mu} = \mu\_0 \begin{pmatrix} \mu\_{rr} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mu\_{r\theta} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mu\_{rz} \end{pmatrix} = \mu\_0 \begin{pmatrix} \mu\_{rt} & \mathbf{0} \\ \mathbf{0} & \mu\_{rz} \end{pmatrix}. \tag{3}
$$

and

$$
\overline{\varepsilon} = \varepsilon\_0 \begin{pmatrix} \varepsilon\_{rr} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \varepsilon\_{r\theta} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \varepsilon\_{rz} \end{pmatrix} = \varepsilon\_0 \begin{pmatrix} \varepsilon\_{rt} & \mathbf{0} \\ \mathbf{0} & \varepsilon\_{rz} \end{pmatrix} \tag{4}
$$

Let consider a circular waveguide of radius R completely filled with anisotropic metamaterial without losses, as represented in the **Figure 1**. The wall of the guide is perfect conductor.

By considering the propagation in the Oz direction and manipulating Eqs. (1) and (2), we obtain the expressions of the transverse electromagnetic fields according to the longitudinal fields.

$$E\_r = \frac{-j}{K\_{c.r}^2} \left( k\_x \frac{\partial E\_x}{\partial r} + \frac{a\mu\_0 \mu\_{r\theta}}{r} \frac{\partial H\_x}{\partial \theta} \right) \tag{5}$$

$$E\_{\theta} = \frac{j}{K\_{c.\theta}^2} \left( \frac{-k\_x}{r} \frac{\partial E\_x}{\partial \theta} + a\mu\_0 \mu\_{rr} \frac{\partial H\_x}{\partial r} \right) \tag{6}$$

$$H\_r = \frac{-j}{K\_{c.\theta}^2} \left( -\frac{\alpha \varepsilon\_0 \varepsilon\_{r\theta}}{r} \frac{\partial E\_x}{\partial \theta} + k\_x \frac{\partial H\_x}{\partial r} \right) \tag{7}$$

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

**Figure 1.** *Geometry of circular waveguide filled with metamaterial.*

$$H\_{\theta} = \frac{j}{K\_{c\tau}^2} \left( -\alpha \varepsilon\_0 \varepsilon\_{rr} \frac{\partial E\_x}{\partial r} - \frac{k\_x}{r} \frac{\partial H\_x}{\partial \theta} \right) \tag{8}$$

with

structure, the propagation modes are exploited. The effects of anisotropic parameter on cutoff frequencies and dispersion characteristics are discussed. Below the cutoff frequency, the back backward waves can propagate in an anisotropic material. The numerical results with our MATLAB code for TM and TE modes were compared to theoretical predictions. Good agreements have been obtained. We analyzed a waveguide filters filled with partially anisotropic metamaterial using the mode matching (MM) technique based on the Scattering Matrix Approach (SMA) which, from the decomposition of the modal fields, are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. We

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

In the anisotropic diagonal metamaterials medium, the Maxwell equations are

¼ �*jωμ:H* !

> ¼ *jωε:E* !

> > 1

1

Let consider a circular waveguide of radius R completely filled with anisotropic metamaterial without losses, as represented in the **Figure 1**. The wall of the guide is

By considering the propagation in the Oz direction and manipulating Eqs. (1)

*∂Ez*

� *ωε*0*εr<sup>θ</sup> r*

and (2), we obtain the expressions of the transverse electromagnetic fields

*kz ∂Ez ∂r*

�*kz r*

CA <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup>

<sup>þ</sup> *ωμ*0*μr<sup>θ</sup> r*

*<sup>∂</sup><sup>θ</sup>* <sup>þ</sup> *ωμ*0*μrr*

*∂Ez <sup>∂</sup><sup>θ</sup>* <sup>þ</sup> *kz*

� �

� �

� �

CA <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

*μrt* 0 0 *μrz* � �

*εrt* 0 0 *εrz* � �

> *∂Hz ∂θ*

> > *∂Hz ∂r*

*∂Hz ∂r*

(1)

(2)

(4)

(5)

(6)

(7)

*:* (3)

extended the application of MM technique to the anisotropic material. This formulation can be a useful tool for engineers of microwave. The metamaterial is largely applied by information technology industries, particularly in the radio frequency devices and microwaves such as the waveguide antennas, the

> ∇ ! � *E* !

> > ∇ ! � *H* !

*μrr* 0 0 0 *μr<sup>θ</sup>* 0 0 0 *μrz*

> *εrr* 0 0 0 *εr<sup>θ</sup>* 0 0 0 *εrz*

*μ* ¼ *μ*<sup>0</sup>

*ε* ¼ *ε*<sup>0</sup>

0

B@

0

B@

*Er* <sup>¼</sup> �*<sup>j</sup> K*2 *c:r*

*<sup>E</sup><sup>θ</sup>* <sup>¼</sup> *<sup>j</sup> K*2 *c:θ*

*Hr* <sup>¼</sup> �*<sup>j</sup> K*2 *c:θ*

patch antennas, the circulators, the resonators and the filters.

**2. Formulation**

expressed as follows

with

and

perfect conductor.

**66**

according to the longitudinal fields.

$$K\_{c.r}^2 = k\_0^2 \varepsilon\_{rr} \mu\_{r\theta} - k\_x^2 \tag{9}$$

$$K\_{c.\theta}^2 = k\_0^2 \varepsilon\_{r\theta} \mu\_{rr} - k\_x^2 \tag{10}$$

$$k\_0^2 = \alpha^2 \varepsilon\_0 \mu\_0 \tag{11}$$

When *E* and *H* are the electric and magnetic field respectively. *ε* and *μ* are the permittivity and permeability. *kz* is the propagation constant in z-direction.

In this chapter, we study rigorously the TE and TM modes in this anisotropic waveguide.

#### **2.1 Transverse electric (TE) modes**

From Eq. (1), the differential equation for z-component can be obtained as follows

$$\frac{\partial^2 H\_x}{\partial r^2} + \frac{\mathbf{1}}{r} \frac{\partial H\_x}{\partial r} + \left(\frac{K\_{c,\theta}^{(h)} \cdot \sqrt{\mu\_{r\theta}}}{K\_{c,r}^{(h)} \cdot \sqrt{\mu\_{rr}}}\right)^2 \frac{\mathbf{1}}{r^2} \frac{\partial^2 H\_x}{\partial \theta^2} + \left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c,\theta}^{(h)}\right)^2 H\_x = \mathbf{0}. \tag{12}$$

Using the separation of the variables (*r,θ*), the expression of the longitudinal magnetic field *Hz* for the *TEmn* modes in the circular metallic waveguide completely filled with anisotropic metamaterial is necessary for the resolution of the differential Eq. (12). *Hz* can be written as follows

$$H\_x^{(h)} = H\_0 \sin\left(\frac{K\_{c.\theta}^{(h)} \cdot \sqrt{\mu\_{r\theta}}}{K\_{c.r}^{(h)} \cdot \sqrt{\mu\_{rr}}} n.\theta\right) J\_n\left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c.\theta}^{(h)} \cdot r\right) e^{-jk\_x x} \tag{13}$$

*Jn* is the Bessel function of the first kind of order *n* (*n* = 0, 1, 2, 3, … ). The expressions (5)–(8) become

$$E\_r^{(h)} = \frac{-j\omega\mu\_0\mu\_{r\theta}}{K\_{c.r}^2r} \left(\frac{K\_{c.\theta}^{(h)}\cdot\sqrt{\mu\_{r\theta}}}{K\_{c.r}^{(h)}\cdot\sqrt{\mu\_{r\theta}}}n.\theta\right)H\_0.\cos\left(\frac{K\_{c.\theta}^{(h)}\cdot\sqrt{\mu\_{r\theta}}}{K\_{c.r}^{(h)}\cdot\sqrt{\mu\_{r\theta}}}n.\theta\right)I\_n\left(\frac{\sqrt{\mu\_{r\theta}}}{\sqrt{\mu\_{r\theta}}}K\_{c.\theta}^{(h)}\cdot r\right)e^{-jk\_rx}\tag{14}$$

$$E\_{\theta}^{(h)} = \frac{j\alpha\mu\_{0}\mu\_{rr}}{K\_{c\theta}^{2}} \left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c\theta}^{(h)}\right) H\_{0} \cdot \sin\left(\frac{K\_{c\theta}^{(h)} \cdot \sqrt{\mu\_{r\theta}}}{K\_{c\tau}^{(h)} \cdot \sqrt{\mu\_{rr}}} n.\theta\right) I\_{n}' \left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c\theta}^{(h)} \cdot r\right) e^{-jk\_{x}x} \tag{15}$$

$$H\_r^{(h)} = \frac{-jk\_\pi}{K\_{c.\theta}^2} \left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c.\theta}^{(h)}\right) H\_0.\sin\left(\frac{K\_{c.\theta}^{(h)}\cdot\sqrt{\mu\_{r\theta}}}{K\_{c.r}^{(h)}\cdot\sqrt{\mu\_{rr}}} n.\theta\right) I\_n'\left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c.\theta}^{(h)}\, r\right) e^{-jk\_\pi x} \tag{16}$$

$$H\_{\theta}^{(h)} = \frac{-jk\_{x}}{K\_{c.r.}^{2}r} \left(\frac{K\_{c.\theta}^{(h)}\sqrt{\mu\_{r\theta}}}{K\_{c.r.}^{(h)}\sqrt{\mu\_{r\}}}n\right)H\_{0.}\cos\left(\frac{K\_{c.\theta}^{(h)}\sqrt{\mu\_{r\theta}}}{K\_{c.r.}^{(h)}\sqrt{\mu\_{r\}}}n.\theta\right)I\_{n}\left(\frac{\sqrt{\mu\_{r\}}}{\sqrt{\mu\_{r\}}}K\_{c.\theta}^{(h)}.r\right)e^{-jk\_{x}x} \tag{17}$$

With *J* 0 *<sup>n</sup>* is the derivative of the Bessel function of the first kind of order n (n = 0, 1, 2, 3, … ).

The boundary conditions are written as follows:

$$E\_{\theta}(r=R) = E\_{x}(r=R) = \mathbf{0} \tag{18}$$

Finally, the propagation constant in TE mode is given by

*k*2

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities…*

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*μrz*

*u*0 *nm R* � �<sup>2</sup>

*nm R*

� �*:* (27)

*<sup>r</sup>*,*eff* ¼ *μrr*, (28)

� �<sup>2</sup> !*:* (29)

(26)

<sup>0</sup>*εr<sup>θ</sup>:μrr* � *<sup>μ</sup>rr*

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ε</sup>rθμrz* j j <sup>p</sup> *: <sup>u</sup>*<sup>0</sup>

We can introduce the following effective permeability and effective permittivity

*εrθμrzk*<sup>2</sup> 0 *: <sup>u</sup>*<sup>0</sup> *nm R*

*<sup>r</sup>*,*eff* >0 and *εTE*

*<sup>r</sup>*,*eff* <0*:*

*<sup>r</sup>*,*eff* < 0 and *εTE*

*<sup>r</sup>*,*eff* >0;

*<sup>r</sup>*,*eff* depends on the sign of *μrz*. In the following, we will consider all

<sup>¼</sup> j j *<sup>ε</sup>r<sup>θ</sup>* <sup>1</sup> � *<sup>f</sup>*

@

1 *<sup>ε</sup>r<sup>θ</sup> <sup>μ</sup>rz* j j*k*<sup>2</sup> 0 *: <sup>u</sup>*<sup>0</sup> *nm R*

*TE c:nm f*

� �<sup>2</sup> ! <sup>&</sup>lt;0*:* (31)

*<sup>r</sup>*,*eff* < 0 below the cutoff frequency whenever

1

A <0, if *f* < *f*

*TE c:nm*

(30)

!<sup>2</sup> 0

*<sup>r</sup>*,*eff* < 0;

to describe the propagation characteristics of the waveguide modes [6, 7, 13].

*μTE*

*<sup>r</sup>*,*eff* <sup>¼</sup> *<sup>ε</sup>r<sup>θ</sup>* <sup>1</sup> � <sup>1</sup>

>0, for *μTE*

<0, for *μTE*

, for *μTE*

*<sup>r</sup>*,*eff :εTE*

*k*ð Þ *TE z:nm* ¼ �

> *f* ð Þ *TE <sup>c</sup>:nm* <sup>¼</sup> *<sup>c</sup>* 2*π*

*εTE*

The cutoff frequency is written

*DOI: http://dx.doi.org/10.5772/intechopen.91645*

Further, it is apparent that:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTE <sup>r</sup>*,*eff :εTE <sup>r</sup>*,*eff* <sup>q</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTE <sup>r</sup>*,*eff :εTE <sup>r</sup>*,*eff* <sup>q</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *μTE <sup>r</sup>*,*eff :εTE <sup>r</sup>*,*eff* <sup>q</sup>

cases that arise from the different sign of *μrz*.

*<sup>ε</sup>r<sup>θ</sup> <sup>μ</sup>rz* j j*k*<sup>2</sup> 0 *: <sup>u</sup>*<sup>0</sup> *nm R*

*εTE*

It can be seen that *μrz* >0 leads to *εTE*

� �2 !

*<sup>r</sup>*,*eff* is rewritten as

*<sup>r</sup>*,*eff* ¼ �j j *εr<sup>θ</sup>* 1 þ

• *kTE <sup>z</sup>* ¼ *k*<sup>0</sup>

• *kTE*

• *kTE*

*εTE*

*<sup>z</sup>* ¼ �*k*<sup>0</sup>

*<sup>z</sup>* ¼ �*jk*<sup>0</sup>

The sign of *εTE*

*2.1.1 First case* μrz > 0

For *εr<sup>θ</sup>* >0, we have.

*<sup>r</sup>*,*eff* <sup>¼</sup> j j *<sup>ε</sup>r<sup>θ</sup>* <sup>1</sup> � <sup>1</sup>

And for *εr<sup>θ</sup>* <0, *εTE*

*εr<sup>θ</sup>* >0 or *εr<sup>θ</sup>* <0.

**69**

Consequently, from Eq. (15), we obtain

$$J\_n' \left(\frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c.\theta}^{(h)}.R\right) = \mathbf{0} \tag{19}$$

This implies

$$
\mu\_{nm}' = \frac{\sqrt{\mu\_{rz}}}{\sqrt{\mu\_{rr}}} K\_{c.\theta}^{(h)}.R \tag{20}
$$

Where *u*<sup>0</sup> *nm* represents the mth zero (m = 1, 2, 3, … ) of the derivative of the Bessel function *J* 0 *<sup>n</sup>* of the first kind of order n.

The constant *H*<sup>0</sup> is determined by normalizing the power flow down the circular guide.

$$P^{\rm TE} = \bigcap\_{0=0}^{R} \left( E\_r^{(h)} H\_\theta^{\* (h)} - E\_\theta^{(h)} H\_r^{\* (h)} \right) r dr d\theta = 1 \tag{21}$$

Where <sup>∗</sup> indicates the complex conjugate. Eq. (21) gives

$$H\_0 = \frac{K\_{c.r}^3}{\sqrt{\alpha \mu\_0 k\_x}} \frac{\sqrt{\mu\_{rz}}}{\mu\_{r\theta}} N\_{nm}^{(h)} \tag{22}$$

With

$$N\_{nm}^{(h)} = \frac{1}{\sqrt{\frac{\sigma\_n}{2}} \left( \left(u\_{nm}'\right)^2 - n^2\right)^{1/2} J\_n\left(u\_{nm}'\right)}\tag{23}$$

$$\sigma\_n = \begin{cases} 2\pi, & \text{if } n = 0\\ \pi - \frac{\sin\left(4\pi a.\mathbf{n}\right)}{4\mathbf{a}.\mathbf{n}}, & \text{if } n > 0 \end{cases} \tag{24}$$

$$a = \frac{K\_{c.\theta}^{(h)} \cdot \sqrt{\mu\_{r\theta}}}{K\_{c.r}^{(h)} \cdot \sqrt{\mu\_{rr}}} \tag{25}$$

*Rigorous Analysis of the Propagation in Metallic Circular Waveguide with Discontinuities… DOI: http://dx.doi.org/10.5772/intechopen.91645*

Finally, the propagation constant in TE mode is given by

$$k\_{x.nm}^{(TE)} = \pm \sqrt{k\_0^2 \varepsilon\_{r\theta} \mu\_{rr} - \frac{\mu\_{rr}}{\mu\_{rx}} \left(\frac{u\_{nm}'}{R}\right)^2} \tag{26}$$

The cutoff frequency is written

*E*ð Þ *<sup>h</sup>*

*H*ð Þ *<sup>h</sup>*

*H*ð Þ *<sup>h</sup>*

With *J* 0

1, 2, 3, … ).

*<sup>θ</sup>* <sup>¼</sup> *<sup>j</sup>ωμ*0*μrr K*2 *c:θ*

> *<sup>r</sup>* <sup>¼</sup> �*jkz K*2 *c:θ*

*<sup>θ</sup>* <sup>¼</sup> �*jkz K*2 *<sup>c</sup>:r:r*

This implies

Where *u*<sup>0</sup>

Bessel function *J*

Eq. (21) gives

With

**68**

guide.

0

ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

!

Consequently, from Eq. (15), we obtain

*<sup>P</sup>TE* <sup>¼</sup> ð *R*

Where <sup>∗</sup> indicates the complex conjugate.

*N*ð Þ *<sup>h</sup>*

*σ<sup>n</sup>* ¼

8 < :

0

2 ð*π*

0

*E*ð Þ *<sup>h</sup> <sup>r</sup> <sup>H</sup>*<sup>∗</sup> ð Þ *<sup>h</sup>*

*<sup>H</sup>*<sup>0</sup> <sup>¼</sup> *<sup>K</sup>*<sup>3</sup>

*nm* <sup>¼</sup> <sup>1</sup> ffiffiffi *σn* 2 p *: u*<sup>0</sup>

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *μrr* p

*K*ð Þ *<sup>h</sup> c:θ* � �

*K*ð Þ *<sup>h</sup> c:θ* � �

*n*

The boundary conditions are written as follows:

*H*0*:*sin

*H*0*:*sin

*H*0*:* cos

*J* 0 *n*

> *u*0 *nm* ¼

*<sup>n</sup>* of the first kind of order n.

ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :R* � �

> ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

*K*ð Þ *<sup>h</sup>*

*nm* represents the mth zero (m = 1, 2, 3, … ) of the derivative of the

*<sup>θ</sup> H*<sup>∗</sup> ð Þ *<sup>h</sup> <sup>r</sup>*

ffiffiffiffiffiffi *μrz* p *μrθ*

*N*ð Þ *<sup>h</sup>*

*:Jn u*<sup>0</sup> *nm* � �

The constant *H*<sup>0</sup> is determined by normalizing the power flow down the circular

*<sup>θ</sup>* � *<sup>E</sup>*ð Þ *<sup>h</sup>*

*c:r* ffiffiffiffiffiffiffiffiffiffiffiffi *ωμ*0*kz* p

*nm* � �<sup>2</sup> � *<sup>n</sup>*<sup>2</sup> � �<sup>1</sup>*=*<sup>2</sup>

*<sup>π</sup>* � sin 4ð Þ *<sup>π</sup>a:*<sup>n</sup>

*<sup>a</sup>* <sup>¼</sup> *<sup>K</sup>*ð Þ *<sup>h</sup>*

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *μrr* p

2*π*, *if n* ¼ 0

*<sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

4a*:*<sup>n</sup> , *if n* <sup>&</sup>gt;<sup>0</sup>

� �

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

!

!

!

*<sup>n</sup>* is the derivative of the Bessel function of the first kind of order n (n = 0,

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *μrr* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *μrr* p

> *K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :* ffiffiffiffiffiffi *μrθ* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>r</sup> :* ffiffiffiffiffiffi *μrr* p

*n:θ*

*n:θ*

*n:θ*

*J* 0 *n*

*J* 0 *n*

*Jn*

*Eθ*ð Þ¼ *r* ¼ *R Ez*ð Þ¼ *r* ¼ *R* 0 (18)

ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

> ffiffiffiffiffiffi *μrz* p ffiffiffiffiffiffi *μrr* p

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :r* � �

*K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :r* � �

> *K*ð Þ *<sup>h</sup> <sup>c</sup>:<sup>θ</sup> :r* � �

¼ 0 (19)

*<sup>c</sup>:<sup>θ</sup> :R* (20)

*rdrdθ* ¼ 1 (21)

*nm* (22)

(23)

(24)

(25)

*e*

*e*

*e*

�*jkzz* (15)

�*jkzz* (16)

�*jkzz* (17)

$$f\_{c.m}^{(TE)} = \frac{c}{2\pi} \frac{1}{\sqrt{|\varepsilon\_{r\theta}\mu\_{rz}|}} \cdot \left(\frac{u\_{nm}'}{R}\right). \tag{27}$$

We can introduce the following effective permeability and effective permittivity to describe the propagation characteristics of the waveguide modes [6, 7, 13].

$$
\mu\_{r, \text{eff}}^{\text{TE}} = \mu\_{rr}, \tag{28}
$$

$$
\varepsilon\_{r,eff}^{TE} = \varepsilon\_{r\theta} \left( 1 - \frac{1}{\varepsilon\_{r\theta} \mu\_{rx} k\_0^2} \cdot \left( \frac{u\_{nm}'}{R} \right)^2 \right). \tag{29}
$$

Further, it is apparent that:

$$\begin{aligned} \bullet \ k\_x^{TE} &= k\_0 \sqrt{\mu\_{r, \text{eff}}^{TE} \varepsilon\_{r, \text{eff}}^{TE}} > 0, \text{ for } \mu\_{r, \text{eff}}^{TE} > 0 \text{ and } \varepsilon\_{r, \text{eff}}^{TE} > 0; \\\\ \bullet \ k\_x^{TE} &= -k\_0 \sqrt{\mu\_{r, \text{eff}}^{TE} \varepsilon\_{r, \text{eff}}^{TE}} < 0, \text{ for } \mu\_{r, \text{eff}}^{TE} < 0 \text{ and } \varepsilon\_{r, \text{eff}}^{TE} < 0; \\\\ \bullet \ k\_x^{TE} &= \pm jk\_0 \sqrt{\mu\_{r, \text{eff}}^{TE} \varepsilon\_{r, \text{eff}}^{TE}}, \text{ for } \mu\_{r, \text{eff}}^{TE} . \varepsilon\_{r, \text{eff}}^{TE} < 0. \end{aligned}$$

The sign of *εTE <sup>r</sup>*,*eff* depends on the sign of *μrz*. In the following, we will consider all cases that arise from the different sign of *μrz*.
